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Cation-anion radius ratio

Ratio rule and stability[edit]

The radius ratio rule defines a critical radius ratio for different crystal structures, based on their coordination geometry.^[1] The idea is that the anions and cations can be treated as incompressible spheres, meaning the crystal structure can be seen as a kind of unequal sphere packing. The allowed size of the cation for a given structure is determined by the critical radius ratio.^[2] If the cation is too small, then it will attract the anions into each other and they will collide hence the compound will be unstable due to anion-anion repulsion; this occurs when the radius ratio drops below the critical radius ratio for that particular structure. At the stability limit the cation is touching all the anions and the anions are just touching at their edges. For radius ratios greater than the critical ratius ratio, the structure is expected to be stable.

The rule is not obeyed for all compounds. By one estimate, the crystal structure can only be guessed about 2/3 of the time.^[3] Errors in prediction are partly due to the fact that real chemical compounds are not purely ionic, they display some covalent character.^[1]

The table below gives the relation between critical radius ratio, ${\displaystyle R}$, and coordination number, ${\displaystyle CN}$, which may be obtained from a simple geometrical proof.^[4]

${\displaystyle R={\sqrt {\tfrac {12}{12-CN}}}{-1}}$

Critical Radius Ratio	Coordination number	Type of void	Crystal structure	Examples
0.1547	3	Trigonal planar	α-B2O3 structure	B2O3
0.2247	4	Tetrahedral	Zincblende structure	ZnS, CuCl
0.4142	6	Octahedral	Rock salt structure	NaCl, MgO
0.7320	8	Cubic	CsCl structure	CsCl, NH4Br

History[edit]

The radius ratio rule was first proposed by Gustav F. Hüttig in 1920.^[5][6] In 1926, Victor Goldschmidt^[5] extended the use to ionic lattices.^[7][8][9] In 1929, the rule was incorporated as the first of Pauling's rules for crystal structures.^[10]

See also[edit]

• Goldschmidt tolerance factor
• Pauling's rules
• Cubic crystal system
• Sphere packing

References[edit]

^ Toofan, Jahansooz (1994). "A Simple Expression between Critical Radius Ratio and Coordination Number". Journal of Chemical Education. 71 (2). American Chemical Society (ACS): 147. doi:10.1021/ed071p147. ISSN 0021-9584. (and Erratum 71(9): 749 doi:10.1021/ed071p749), Following the erratum, equations should read ${\displaystyle \alpha =\arcsin {({\tfrac {12-CN}{12}})^{\tfrac {1}{2}}}}$ and ${\displaystyle R=({\tfrac {12}{12-CN}})^{\tfrac {1}{2}}{-1}}$.